\documentclass{article}
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\begin{document}
$1 - p$
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$p$
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$1 - pP$
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$ p_x^{(n)} = P(X = x) = {n \choose x} \cdot p^x \cdot q^{n-x} $
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$n$
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$x$
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$p_x^{(n)}$
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$q = 1 -p$
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$ {n \choose x} = \frac{n \cdot (n-1) \cdot (n-2) \cdot \dots \cdot (n-x+1)}{1 \cdot 2 \cdot \dots \cdot x} = \prod_{i=1}^x \frac{n+1-i}{i} $
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${n \choose x}$
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$ tt! = tt \cdot (tt - 1) \cdot (tt - 2) \cdot \dots \cdot 2 \cdot 1 = \prod_{i=1}^{tt} i $
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$tt!$
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$p_x^{(n)} = P(X = x) = {n \choose x} \cdot p^x \cdot q^{n-x}$
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$q = 1 - p$
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$n = 8$
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$p = 0.3$
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$p = 0.5$
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$ f(x) = \frac{1}{\pi (1 + x^2)} $
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$ f(x) = ( p \cdot (1 - p)^{|x|} ) / (2 - p) $
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$p = 0.2$
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$p = 0.8$
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$rn$
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$\leq rn <$
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$p = \frac{1}{\mbox{interval length}}$
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$[$
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$lo \leq rn < hi$
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$\mu$
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$\sigma^2$
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$k$
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$k = \frac{\mu^2}{\sigma^2+0.5}$
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$\alpha$
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$\alpha = \frac{k}{\mu}$
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$\geq 1$
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$> 0$
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$k = 1$
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$\alpha = 0.5$
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$\mu \leq 0$
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$k = 1,2,3,...$
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$k=1,2,3$
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$ f(x) = (k \cdot \alpha)^k \cdot x^{(k - 1)} \cdot e^{-k \alpha x} $
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$\alpha > 0$
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$k > 0$
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$k = 1, 2, 3$
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$(1-p)$
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$x-1$
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$x > 0$
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$ f(x) = p \cdot (1-p)^{x-1} $
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$x = 0$
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$f(x) = 0$
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$p = \frac{N_1}{N}$
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$p = \frac{1}{2} \cdot (1 - \sqrt{\frac{\sigma^2 / \mu^2 - 1}{\sigma^2 / \mu^2 + 1}})$
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$\geq 0$
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$p = 0.$
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$T_1$
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$N_1$
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$T_2$
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$N_2$
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$N = N_1 + N_2$
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$ P(k) = {pN \choose k} {N(1 - p) \choose n - k} / {N \choose k} $
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$p \cdot N$
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$N \cdot (1 - p)$
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$P(k)$
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$n \leq N$
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$\sqrt{e}$
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$e(e-1)$
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$V(X)=\sigma^2$
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$x \leq 0$
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$ f_{log}(x) = \frac{1}{\sqrt{2 \pi \sigma^2 x}} \cdot exp \left(- \frac{(ln x - \mu)^2}{2 \sigma^2}\right) $
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$f_n(x)$
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$f_{log}(x) = \frac{1}{x}f_n(ln x)$
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$\sigma$
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$V(x) = \sigma^2$
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$\mbox{mean} = \sqrt{e}$
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$\mbox{standard deviation} = e(e-1)$
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$\lambda$
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$\lambda = $
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$x < 0$
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$ f(x) = \lambda \cdot e^{- \lambda x} $
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$x \geq 0$
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$\lambda > 0$
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$\lambda = 1.0$
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$\lambda = 2.0$
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$\lambda = 4.0$
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$\lambda = \mbox{mean}$
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$pStdDev = \sqrt{\mbox{variance}}$
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$ f(x) = \frac{1}{\sqrt{2 \pi \sigma^2}} \cdot exp \left(- \frac{(x- \mu)^2}{2 \sigma^2}\right) $
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$0$
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$1$
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$\mu = 0$
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$\sigma = 1$
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$V(X)$
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$ V(X) = \sigma^2 $
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$\sigma = \sqrt{V(X)}$
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$ p_x^{(n)} = P(X = x) = ( \lambda^x \cdot e^{- \lambda} ) / x! $
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$\lambda = n \cdot p$
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$\lambda = 5.0$
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$F(X)$
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$p = \{p(x): x \in \chi\}$
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$H(p)$
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$ H(p) = - \sum_{x \in \chi} p(x) \cdot \log {p(x)} = E_p [- \log {p(X)}] $
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$q = \{q(x): x \in \chi\}$
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$q$
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$ D(p,q) = \sum_{x \in \chi} p(x) \cdot \log {\frac{p(x)}{q(x)}} = E_p \big[ \log {\frac{p(X)}{q(X)}}\big] = E_p [ \mbox{log likelihood ratio} ] $
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$l$
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$ l = \sum_{i=1}^k \log {P( x_i )} $
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$P( x_i )$
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$x_i$
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$ n_i \equiv ( a \cdot n_{i-1} ) \pmod m $
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$n_i$
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$m$
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$n_0$
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$a$
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$ \mbox{\ }\\ \noindent n_{1,i} \equiv ( a_1 \cdot n_{1, i-1} ) \pmod {m_1}\\ n_{2,i} \equiv ( a_2 \cdot n_{2, i-1} ) \pmod {m_2}\\ n_{3,i} \equiv ( a_3 \cdot n_{3, i-1} ) \pmod {m_3} $
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$U_i$
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$ U_i \equiv (\frac{n_{1,i}}{m_1} + \frac{n_{2,i}}{m_2} + \frac{n_{3,i}}{m_3}) \pmod {1.0} $
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$a_i$
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$m_i$
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$i = 1, 2, 3$
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$6953607871644$
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$z_i$
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$F$
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$ x_i = F^{-1}(z_i) $
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$n_{0,1},\mbox{\ }n_{0,2},\mbox{\ }n_{0,3}$
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$n_{i,1},\mbox{\ }n_{i,2},\mbox{\ }n_{i,3}$
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$n_{i,1}$
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$n_{i,2}$
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$n_{i,3}$
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$\beta$
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$alpha$
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$ f(x) = \frac{\alpha}{\beta} \cdot e^{-\frac{1}{\beta} \cdot x^\alpha} \cdot x^{\alpha - 1} $
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$\alpha = 1\mbox{,\ } \beta = 2$
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$\alpha = 2\mbox{,\ } \beta = 1$
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$\alpha = 2\mbox{,\ } \beta = 3$
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\end{document}
